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Exploring Roots of Quadratics

Erin Mueller

 

 

 

The general quadratic has equation . All of the possible roots (real and imaginary) of this polynomial can be found using the quadratic formula , in which the radicand  can tell us which roots are real and which are imaginary. Lets explore how the variable b changes the polynomial and the resulting roots. Below is a graph of a quadratic where b is the altering variable.

 

 

As you can see, the vertex is traveling along a path that looks much like another parabola itself. The vertex travels through the bottom two quadrants depending on the sign of b. If b is positive, the vertex travels to the right and in this case, parabola will be on the positive side of the x-axis. If b is negative, the vertex travels to the left and will be on the negative side of the x-axis.

 

Now lets consider graphing the parabola in the x-b plane. This means that our b value will become our dependent variable while our x value remains the independent variable.

 

By solving the general parabola for b, we will obtain a rational equation.

 

 

 

 

 

 

When c=1, our graph of this equation looks like this:

 

 

 

From the above graph, we can conclude that when c=1, the range of the function includes all real numbers except when . We can also see that whenever our x-value is negative, our y-value (our actual b) is also negative and vice versa when our x-value is positive.

 

If we take any particular value of b, say b = 3, and overlay this equation on the graph we add a line parallel to the x-axis. If it intersects the curve in the x-b plane the intersection points correspond to the roots of the original equation for that value of b. We have the following graph.

 

 

 

 

The graph above shows that when b=3 in our original quadratic equation , the roots for this equation are approximately x=-.4 and x=-2.6. From the above, we have now graphed our equation in x-b plane. This means that instead of looking at x=intercepts for our roots, we are now looking at the intersections of our rational polynomial with the horizontal line b=3.

 

Now lets look at this graph in motion.

 

 

 

 

    For each value of b we select, we get a horizontal line. It is clear on a single graph that we get two negative real roots of the original equation when b > 2, one negative real root when b = 2, no real roots for -2 < b < 2, One positive real root when b = -2, and two positive real roots when b < -2.

 

    Consider the case when c = - 1 rather than + 1.

 

 

As you can see, we now have one positive root at approximately x=.1 and another at approximately x=-7.9. Lets try other values of c.

 

How about c=0?

 

 

 

 

When c=0, we only have one real negative root. Since we still have a quadratic, we must have two roots. In this case, one is real and one is imaginary.

 

 

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